Attached objects rotate at different pivot point on rig?
There are two solutions that I can think of. 

Solution 1: 

Make the head and the mustache one mesh. You can do that by selecting both and pressing Ctrl+J. Now you have to make sure that the pivot point is stil right. If not select the point where the head connects with the body and move your 3D-Cursor to it by pressing Shift+S and selecting "Cursor to selected". Now press Ctrl+Shift+Alt+C and select "Origin to 3D Cursor". 

Solution 2: 

If you want to move the head/mustache only once or twice you can move your 3D Cursor to the point where the head connects with the body as described above. Then just change the pivot center to "3D Cursor" (see screenshot). You don't need to make the head and the mustache one mesh in this option.